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COMPARTMENTAL ANALYSIS REVISITED
Pharmacological Research, Vol. 39, No. 6, 1999 Article No. phrs.1999.0467, available online at http:rrwww.idealibrary.com on
COMPARTMENTAL ANALYSIS REVISITED ALDO RESCIGNOU College of Pharmacy, Uni¨ ersity of Minnesota, Minneapolis, MN 55082, USA Accepted 13 January 1999
This paper gives a precise definition of compartment and shows that even when a compartmental model is not exactly identifiable, a range of its pharmacokinetic parameters can be determined. It is thus better to know approximate values of clearly defined parameters than exact values of quantities with no physical or physiological content. Q 1999 Academic Press KEY
WORDS:
pharmacokinetics, compartment, turnover time, transfer time, dilution factor.
INTRODUCTION
the compartment, and r Ž t . the rate of entry into the compartment.
Compartmental analysis dates back to Teorell w1, 2x and to Artom et al. w3x. Since then, thousands of papers have been written on this subject, and a large number of computer programs have been made available to implement it. Nevertheless the theoretical foundations of this method are not always clear, and its results are frequently misinterpreted. In this paper, I shall briefly comment on some of the basic assumptions of compartmental analysis and on some of its possible applications.
DEFINITION OF COMPARTMENT Many different definitions of compartment can be found in the literature; the simplest one is: ‘A compartment is a set of particles defined by a physical boundary and having identical kinetic properties’ w4, 5x. By accepting this definition, expressions like linear compartments, homogeneous compartment and well-stirred compartment become redundant. In this paper, I plan to restrict myself to compartments as defined above; therefore I shall consider only kinetics of order one with a constant rate w6x. With these hypotheses, if x Ž t . is the number of particles in a compartment, the differential equation d xrdt s yK x Ž t . q r Ž t .
Ž1.
holds, with K the constant rate of elimination from U
Corresponding author. 14767 Square Lake Trail N., Stillwater, MN 55082-9278, USA.
1043]6618r99r060471]08r$30.00r0
PURPOSE OF MODELING It has been said ad nauseam that the purpose of modeling is to test hypotheses w7]10x, yet frequently we see in the literature statements like ‘models of data’ w11x, meaning that experimental data are fitted to ‘abstract mathematical functions... that do not necessarily have any physical basis’; the so-called models of data may be useful for synthetically describing data observed in the past or to interpolate some values, but such an analysis has no scientific content per se. A classical example of ‘model of data’ is the Almagest; in it Ptolemy used abstract mathematical functions Žcycles and epicycles. to describe the movements of all known celestial bodies, obtaining a remarkably good fit; but the good fit did not help in interpreting the data or in predicting future observations. On the contrary, Aristarchus of Samos Ž; 310]230 BC. first, then more consistently Copernicus ŽAD 1473]1545., put forward the heliocentric hypothesis, thereby basing their computations not on the mere fitting of data Žin fact Copernicus fitting was not better than Ptolemy’s., but on a physical basis. In short, a compartmental equation like Ž1. implies the hypothesis that the particles present in the compartment leave it at a constant fractional rate, K; this may or may not be the case. There are instances where the particles present in a given space are eliminated with a non-linear process, or Q 1999 Academic Press
Pharmacological Research, Vol. 39, No. 6, 1999
472
with a linear process but not exponentially Žfor instant by diffusion., or by an exponential process with a non-constant fractional rate. Of course all results obtained with the application of equation Ž1., an apparently very general equation, are valid only within the validity of the stated hypotheses. Fitting the experimental data with a sum of exponential functions, by itself, does not guarantee the validity of the above hypotheses, and the determination of the parameters of the exponential functions is not scientifically meaningful if those parameters cannot be given a physical or physiological interpretation. Another example of dubious use of compartmental analysis is in the choice of the model. This choice is sometimes done according to some criteria like ‘identifiability’, as though an identifiable model were more ‘true’ than a non-identifiable one. The truth is that a ‘non-identifiable’ model may permit the determination of the range of some meaningful parameters, while with an ‘identifiable’ model you may compute the exact value of some parameters deprived of any physical or physiological meaning. I shall now try to show, with some examples, how to get the most from compartmental models.
DEGREES OF FREEDOM
A system of n compartments is completely described by nŽ n y 1. transfer rates and n turnover rates, for a total of n2 parameters. If one compartment is sampled after a bolus administration, we observe a sum of n exponential functions, therefore we know n exponents and n y 1 coefficients, for a total of 2 n y 1 parameters; thus a system of n compartments can be determined with Ž n y 1. 2 degrees of freedom. A system of two compartments can be determined with just 1 degree of freedom, with three compartments the degrees of freedom are 4, and with more compartments the number of degrees of freedom increases very quickly. Nevertheless, there is a considerable amount of meaningful information that can be extracted from such systems. This problem has been dealt with previously in some special cases, notably by Berman and Schoenfeld w12x and by Mordenti and Rescigno w13x. Here I shall show a general solution in the case of two and three compartments, leaving the solution for more compartments to the imagination of the reader.
d x1 s yK 1 x 1 q k 21 x 2 dt d x2 s qk 12 x 1 y K 2 x 2 , dt
Ž2.
where x 1Ž t ., x 2 Ž t . are the amount of drug at time t in each compartment, the capital K s are the turnover rates of the compartments, and the small ks are the transfer rates between compartments. Obviously there are the restrictions 0 F k 12 F K 1 , 0 F k 21 F K 2 .
Ž3.
For simplicity we suppose that the dose D of drug is administered into the first compartment as a bolus at time t s 0, and sampled in that same compartment; we thus have the initial conditions x 1Ž 0 . s D, x 2 Ž 0 . s 0. What we measure is not the amount present in a compartment, but its concentration, i.e. c1Ž t .. At this point it is very convenient to manipulate the above equations with the operational notation first developed in 1959 w14x, and in 1972 applied to compartmental analysis w6x. The interested reader may get more information from the two above quoted references. In Appendix A, I have listed only a number of rules as they will be used in the following pages. Equations Ž2. can be written, using rule 1 Žsee Appendix A., s x 1 4 y x 1Ž 0 . s yK 1 x 1 4 q k 21 x 2 4 s x 2 4 y x 2 Ž 0 . s qk 12 x 1 4 y K 2 x 2 4 and incorporating the initial conditions, q Ž s q K 1 . x 1 4 y k 21 x 2 4 s D yk 12 x 1 4 q Ž s q K 2 . x 2 4 s 0.
Ž4.
Using ordinary algebra, from equations Ž4. we have the solution x1 4 s
s q p1 , s 2 q q1 s q q 2
Ž5.
where TWO COMPARTMENTS
The differential equations describing a system of two compartments are
p1 s K 2 ,
q1 s K 1 q K 2 ,
q2 s K 1 K 2 y k 12 k 21 . Ž 6 .
On the other hand, we measure the concentration of the drug, and from the experimental data we get a
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473
sum of two exponential functions c1Ž t . s A1 eyl 1 t q A 2 ey l 2 t ,
Ž7.
istered as a bolus at time t s 0 is given by the integral from 0 to ` of x 1Ž t ., therefore, using rule 3 on equation Ž5., T1 s p1rq2
and using rule 2, c1 4 s s
A1 A2 q s q l1 s q l2 Ž A1 q A 2 . s q A1 l2 q A 2 l1 s 2 q Ž l1 q l 2 . s q l1 l 2
Ž8.
We know that c1Ž0. s DrV1 , where V1 is the volume of the first compartment, therefore from equation Ž6. we get V1 s
D , A1 q A 2
p1 s
A1 l2 q A 2 l1 , A1 q A 2
As shown by Rescigno and Gurpide w16x, the exit time from the sampling compartment is given by the ratio w H0` t ? x 1Ž t . dt xrw H0` x 1Ž t . dt x, therefore using rules 3 and 4, V1 s
Following Mordenti and Rescigno w13x, the exit time from the system of both compartments can be computed with the formula `
V 12 s
q1 s l1 q l2 ,
K 1 s q1 y p 1 ,
K 2 s p1 ,
k 12 k 21 s p1Ž q1 y p1 . y q2 .
k 21 s
p 1 Ž q1 y p 1 . y q 2 a
p1 q a . q2
q1 q 1 y F V 12 F 1 . q2 p1 q2
p 1 Ž q1 y p 1 . y q 2 F p1 , a
q2 F a F q1 y p 1 . p1
, w k 10 x 1Ž t . q k 20 x 2 Ž t .x dt
Using inequalities Ž9.,
The dilution factor u , as shown by Mordenti and Rescigno w13x, is
therefore, q1 y p 1 y
`
V 12 s
and using conditions Ž3., 0 F a F q1 y p 1 , 0 F
t ? w k 10 x 1Ž t . q k 20 x 2 Ž t .x dt
where k 10 s K 1 y k 12 and k 20 s K 2 y k 21 are not known exactly; but we can compute a range for this parameter. Using the arbitrary parameter a and the previous results we get
Introducing the arbitrary parameter a we have k 12 s a ,
H0
H0
q2 s l1 l2 . Now from equation Ž5. we can compute
q1 1 y . q2 p1
us
Ž9.
In summary, K 1 s q1 y p 1 , K 2 s p 1 , q q1 y p1 y 2 F k 12 F q1 y p1 , p1 q2 p1 y F k 21 F p1 . q1 y p 1
DETERMINATION OF OTHER PARAMETERS Besides turnover times and transfer times, there are many other important parameters that can be computed from the data available. It has been shown w13, 15x that the permanence time in the compartment where the drug was admin-
K 2 q k 12 , K2
therefore
us1q
a p1
and q1 q q y 22 F u F 1 . p1 p1 p1
EXAMPLE Suppose we gave an i.v. dose of 100 mg of a drug to a subject, and measured the concentration c1Ž t . s 33.5ey1 .0 t q 10.2ey4.0 t ,
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474
where the coefficients are measured in mg ly1 and the exponents in hy1 . We compute
d x1 s yK 1 x 1 q k 21 x 2 q k 31 x 3 dt d x2 s qk 12 x 1 y K 2 x 2 q k 32 x 3 dt d x3 s qk 13 x 1 q k 23 x 2 y K 3 x 3 dt
100 s 2.29 l 33.5q 10.2 33.5= 4.0q 10.2= 1.0 p1 s s 3.30 hy1 33.5q 10.2 V1 s
q1 s 1.0q 4.0s 5.0 hy1 q2 s 1.0= 4.0s 4.0 h
Ž 10.
with the restrictions
y2
ki j
and
G 0,
i , j s 1,2,3;
Sj ki j F K i , i
s 1,2,3
Ž 11.
y1
K 1 s 5.0y 3.30s 1.7 h K 2 s 3.30 hy1
and the initial conditions x 1Ž 0 . s D,
k 12 k 21 s 3.30= 1.7y 4.0s 1.61 hy2 Now we introduce the parameter a with the restrictions
x 2 Ž 0 . s x 3 Ž 0 . s 0.
Proceeding as before, we have the equations q Ž s q K 1 . x 1 4 y k 21 x 2 4 y k 31 x 3 4 s D
4.0 1.7y F a F 1.7, 3.30
yk 12 x 1 4 q Ž s q K 2 . x 2 4 y k 32 x 3 4 s 0 yk 13 x 1 4 y k 23 x 2 4 q Ž s q K 3 . x 3 4 s 0
therefore 0.49F k 12 F 1.7
and using ordinary algebra,
0.95F k 21 F 3.29 x1 4 s
We can also compute the permanence time in the first compartment, 3.30 T1 s s 0.82 h, 4.0 the exit time from the first compartment, V1 s
5.0 1 y s 0.95 h, 4.0 3.3
the exit time from the system, 5.0 1 5.0 y F V 12 F 4.0 3.3 4.0 thence
s 2 q p1 s q p 2 s 3 q q1 s 2 q q 2 s q q 3
Ž 12.
where p1 s K 2 q K 3 , p 2 s K 2 K 3 y k 23 k 32 , q1 s K 1 q K 2 q K 3 , q2 s K 1 K 2 y k 12 k 21 q K 1 K 3 y k 13 k 31 qK 2 K 3 y k 23 k 32 , q3 s K 1 K 2 K 3 y K 1 k 23 k 32 y K 2 k 13 k 31 y K 3 k 12 k 21 yk 12 k 23 k 31 y k 13 k 32 k 21 .
Ž 13.
Now we can compute 0.95F V 12 F 1.25,
and finally the dilution factor 5.0 4.0 5.0 y FuF 2 3.30 3.30 3.30 thence
K 1 s q1 y p 1 ,
K2 s a ,
K 3 s p1 y a ,
where a is an arbitrary parameter; we are left with equations Ž14.: p 2 s a Ž p1 y a . y k 23 k 32 q2 s p1Ž q1 y p1 . y k 12 k 21 y k 13 k 31 q p 2
1.15F u F 1.52.
q3 s a Ž p1 y a .Ž q1 y p1 . y Ž q1 y p1 . k 23 k 32 Ž 14. ya k 13 k 31 y Ž p1 y a . k 12 k 21
THREE COMPARTMENTS The differential equations describing a system of three compartments are
yk 12 k 23 k 31 y k 13 k 32 k 21 . These are three equations in the six unknown transfer rates; the solution depends on three more arbitrary parameters besides a .
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475
From the first of the above equations we get
then for the products of the transfer rates
k 23 k 32 s a Ž p1 y a . y p 2 , but from Ž11. it is necessary that
k 12 k 21 s
K 2 Ž q2 y K 1 K 2 y K 2 K 3 . y q3 , K3 y K2
k 13 k 31 s
q3 y K 3 Ž q2 y K 1 K 3 y K 2 K 3 . . K3 y K2
k 23 k 32 G 0,
To determine the transfer rates we introduce two arbitrary parameters b and g , such that
therefore
a Ž p1 y a . y p 2 G 0,
k 12 s b , k 21 s k 12 k 21rb , k 13 s g , k 31 s k 13 k 31rg .
and solving this inequality we get p1 y 2
(ž
p1 2
2
/ yp
2
FaF
p1 q 2
p1 2
(ž
2
/ yp
2
.
Conditions Ž11. require that
Ž 15.
b ) 0, g ) 0, b q g F K 1 , k 12 k 21rb F K 2 , k 13 k 31rg F K 3 ,
For a equal to its lowest or largest admissible value we have k 23 k 32 s 0,
therefore
while the largest possible value for k 23 k 32 is given by
as
p1 , 2
k 23 k 32 s
ž p2 / y p . 1
2
b q g F K1 b G k 12 k 21rK 2 g G k 13 k 31rK 3
2
The large number of degrees of freedom in the determination of the transfer rates may seem forbidding, but there is still a large amount of information that we can get from the available data. We start by considering two particular cases, i.e. a catenary system and a mammillary system.
MAMMILLARY SYSTEM
The three above inequalities define the triangle ABC Žsee Fig. 1. whose points correspond to acceptable values of the parameters b and g . Observe that along line AB, b assumes its minimum value, therefore k 21 s K 2 and there is no loss from the second compartment. Along line BC, b q g s K 1 , therefore k 12 q k 13 s K 1 and there is no loss from the first compartment. Along line CA, g assumes its minimum value, therefore k 31 s K 3 and there is no loss from the third compartment. All
In a mammillary system w6x, k 23 s k 32 s 0, therefore equations Ž13. become p1 s K 2 q K 3 , p2 s K 2 K 3 , q1 s K 1 q K 2 q K 3 ,
Ž 16.
q2 s K 1 K 2 y k 12 k 21 q K 1 K 3 y k 13 k 31 q K 2 K 3 , q3 s K 1 K 2 K 3 y K 2 k 13 k 31 y K 3 k 12 k 21 , and we solve immediately for the turnover rates K 1 s q1 y p 1 , K2 s
p1 y 2
K3 s
p1 q 2
(ž (ž
p1 2 p1 2
2
/ yp 2
/
2
,
y p2 ,
Fig. 1. Mammillary model of a three-compartment system. Range of parameters b and g that define physically realizable values of the model parameters.
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476
points inside triangle ABC correspond to intermediate states.
CATENARY SYSTEM In a catenary system w6x, k 13 s k 31 s 0, therefore equations Ž13. become p1 s K 2 q K 3 , p 2 s K 2 K 3 y k 23 k 32 , q1 s K 1 q K 2 q K 3 , q2 s K 1 K 2 y k 12 k 21 q K 1 K 3 q K 2 K 3 y k 23 k 32 , q3 s K 1 K 2 K 3 y K 1 k 23 k 32 y K 3 k 12 k 21 , Fig. 2. Catenary model of a three-compartment system. Range of parameters b and g that define physically realizable values of the model parameters.
and we can compute immediately K 1 s q1 y p 1 while from the remaining four equations K 2 q K 3 s p1 ,
EXAMPLE Suppose we have found that
K 2 K 3 y k 23 k 32 s p 2 , x1 4 s
k 12 k 21 q k 23 k 32 s K 1 K 2 q K 1 K 3 q K 2 K 3 y q2 , K 3 k 12 k 21 q K 1 k 23 k 32 s K 1 K 2 K 3 y q3 , we can compute the turnover rates K 2 and K 3 and the products of the transfer rates k 12 k 21 and k 23 k 32 . To determine the individual transfer rates we introduce the indeterminate parameters b and g , such that
With the hypothesis that the system is mammillary we can compute the turnover rates K 1 s 2.0,
K 2 s 1.0,
k 12 k 21 s 0.60,
k 12 s k 12 k 21rb ,
K 3 s 3.5
and the products
k 21 s b ,
k 13 k 31 s 1.70;
then with the two arbitrary parameters b and g we have
k 23 s g ,
k 12 s b ,
k 32 s k 23 k 32rg ,
k 21 s 0.60rb ,
k 13 s g ,
k 31 s 1.70rg
with the conditions
with the conditions k 12 k 21rb F K 1 ,
s 2 q 4.5s q 3.5 . s 3 q 6.5s 2 q 10.5s q 4.25
b q g F K2 ,
k 23 k 32rg F K 3
The three above inequalities define the triangle ABC Žsee Fig. 2. whose points correspond to acceptable values of the parameters b and g . Observe that along line AB, b assumes its minimum value, therefore k 12 s K 1 and there is no loss from the first compartment. Along line BC, b q g s K 2 , therefore k 21 q k 23 s K 2 and there is no loss from the second compartment. Along line CA, g assumes its minimum value, therefore k 32 s K 3 and there is no loss from the third compartment. All points inside triangle ABC correspond to intermediate states.
b q g F 2.0,
b G 0.60, g G 0.486.
If we accept the alternative hypothesis that the system is catenary, we can compute K 1 s 2.0 and we are left with the four equations K 2 q K 3 s 4.5, K 2 K 3 y k 23 k 32 s 3.5, k 12 k 21 q k 23 k 32 y 2 K 2 y 2 K 3 y K 2 K 3 s y10.5, K 3 k 12 k 21 q 2 k 23 k 32 y 2 K 2 K 3 s y4.25,
Pharmacological Research, Vol. 39, No. 6, 1999
whose solution is K 2 s 3.125,
K 3 s 1.373,
k 12 k 21 s 2.0,
k 23 k 32 s 0.80,
477
in fact when a is in the interval 2.25]3.5, we have the same solution with the second compartment switched with the third one. Eliminating k 13 k 31 from the above equations we have
then we eliminate K 2 q K 3 and we get k 12 k 21 s 2.0,
K 3 s 1.375,
K 2 s 3.125,
and finally k 23 k 32 s 0.80. Now we introduce two parameters b and g , such that k 21 s b ,
k 12 s 2.0rb ,
k 23 s g ,
k 32 s 0.80rg .
k 12 k 21 s
2 Ž 1.375y a . y k 12 k 23 k 31 y k 13 k 32 k 21 ; 2 Ž 2.25y a .
for this expression to be non-negative it is necessary that
a F 1.375. We conclude that 1.0F a F 1.375; our parameters are therefore K 1 s 2, 1.0F K 2 F 1.375, 3.125F K 3 F 3.5, 0 F k 23 k 32 F 0.7969.
Inequalities Ž11. require that
b G 1.0,
b q g F 3.125, g G 0.604.
General case Without any special hypothesis on the transfer rates, equations Ž14. become 3.5s a Ž 4.5y a . y k 23 k 32 ,
Even though we were not able to determine the exact value of the turnover rates, we were able to restrict their range to a reasonably narrow interval. Additionally, we can compute the permanence time in the first compartment,
10.5s 4.5 Ž 6.5y 4.5. y k 12 k 21 y k 13 k 31 q 3.5,
T1 s
4.25s a Ž 4.5y a .Ž 6.5y 4.5. y Ž 6.5y 4.5. k 23 k 32 ya k 13 k 31 y Ž 4.5y a . k 12 k 21
the exit time from the first compartment,
yk 12 k 23 k 31 y k 13 k 32 k 21 ,
V1 s
thence k 23 k 32 s ya 2 q 4.5 a y 3.5,
10.5 4.5 10.5 y F V 12 F 4.25 3.5 4.25 thence
qk 13 k 32 k 21 s y2 a q 9.0 a y 4.25, 2
and finally
10.5 4.5 y s 1.714 h, 3.5 3.5
the exit time from the system,
k 12 k 21 q k 13 k 31 s 2.0, Ž 4.5y a . k 12 k 21 q a k 13 k 31 q 2 k 23 k 32 q k 12 k 23 k 31
3.5 s 0.824 h, 4.25
1.18F V 12 F 2.47, and finally the dilution factor
k 23 k 32 s ya 2 q 4.5 a y 3.5,
10.5 4.25? 4.5 10.5 y FuF 2 3.5 3.5 Ž 3.5.
k 12 k 21 q k 13 k 31 s 2.0, Ž 4.5y a . k 12 k 21 q a k 13 k 31 q k 12 k 23 k 31
thence
qk 13 k 32 k 21 s 2.75. We have three equations with six unknowns; they can be solved with three additional degrees of freedom, besides the parameter a ; we already know, from inequality Ž15., that 1.0F a F 3.5. Now observe that without any loss of generality we can restrict the range of a to the interval 1.0F a F 2.25;
1.44F u F 3.00.
APPENDIX A Rule 1: Derivative of a function
½ ddtf 5 s s f 4 y f Ž0. .
Pharmacological Research, Vol. 39, No. 6, 1999
478
Rule 2: Exponential function aeyl t 4 s
a . sql
Rule 3: Integral of a function `
H0
f Ž t . dt s lim f 4 . sª0
Rule 4: First moment of a function `
d f lim H0 t ? f Ž t .dt s sª0 ds
4
.
REFERENCES 1. Teorell T. Kinetics of distribution of substances administered to the body. I. The extravascular modes of administration. Arch Int Pharmacodyn Ther ´ 1937; 57: 205]25. 2. Teorell T. Kinetics of Distribution of Substances Administered to the Body. II. The Intravascular Modes of Administration. Arch Int Pharmacodyn Ther ´ 1937; 57: 226]40. 3. Artom C, Sarzana G, Segre ´ E. Influence des grasses alimentaires sur la formation des phospholipides dans les tissues animaux Žnouvelles recherches .. Arch Int Physiol 1938; 47: 245]76. 4. Rescigno A, Segre G. La Cinetica dei Farmaci e dei Traccianti Radioatti¨ i. Torino: Boringhieri, 1961.
5. Rescigno A, Segre G. Drug and Tracer Kinetics. Waltham, Massachusetts: Blaisdell, 1966. 6. Rescigno A, Beck JS. Compartments. In: Rosen R, Ed. Foundations of Mathematical Biology. New York: Academic Press, 1972: 255]322. 7. Nooney GC. Mathematical models, reality and results. J Theor Biol 1965; 9: 239]52. 8. Bergner P-EW. Tracer theory: a review. Isot Radiat Technol 1966; 3: 245]62. 9. Beck JS, Rescigno A. Calcium kinetics: the philosophy and practice of science. Phys Med Biol 1970; 15: 566]7. 10. Rescigno A, Beck JS. The use and abuse of models. J Pharmacokinet Biopharm 1987; 15: 327]40. 11. DiStefano JJ, Landaw EM. Multiexponential, multicompartmental, and noncompartmental modeling. I. Methodological limitations and physiological interpretations. Am J Physiol 1984; 246: R651]64. 12. Berman M, Schoenfeld R. Invariants in experimental data on linear kinetics and the formulation of models. J Appl Phys 1956; 27: 1361]70. 13. Mordenti J, Rescigno A. Estimation of permanence time, exit time, dilution factor, and steady-state volume of distribution. Pharm Res 1992; 9: 17]25. 14. Mikusinski J. Operational Calculus. London: Pergamon Press, 1959. 15. Rescigno A, Thakur AK, Brill AB, Mariani G. Tracer kinetics: a proposal for unified symbols and nomenclature. Phys Med Biol 1990; 35: 449]65. 16. Rescigno A, Gurpide E. Estimation of average times of residence, recycle, and interconversion of bloodborne compounds using tracer methods. J Clin Endocrinol Metab 1973; 36: 263]76.
COMPARTMENTAL ANALYSIS REVISITED ALDO RESCIGNOU College of Pharmacy, Uni¨ ersity of Minnesota, Minneapolis, MN 55082, USA Accepted 13 January 1999
This paper gives a precise definition of compartment and shows that even when a compartmental model is not exactly identifiable, a range of its pharmacokinetic parameters can be determined. It is thus better to know approximate values of clearly defined parameters than exact values of quantities with no physical or physiological content. Q 1999 Academic Press KEY
WORDS:
pharmacokinetics, compartment, turnover time, transfer time, dilution factor.
INTRODUCTION
the compartment, and r Ž t . the rate of entry into the compartment.
Compartmental analysis dates back to Teorell w1, 2x and to Artom et al. w3x. Since then, thousands of papers have been written on this subject, and a large number of computer programs have been made available to implement it. Nevertheless the theoretical foundations of this method are not always clear, and its results are frequently misinterpreted. In this paper, I shall briefly comment on some of the basic assumptions of compartmental analysis and on some of its possible applications.
DEFINITION OF COMPARTMENT Many different definitions of compartment can be found in the literature; the simplest one is: ‘A compartment is a set of particles defined by a physical boundary and having identical kinetic properties’ w4, 5x. By accepting this definition, expressions like linear compartments, homogeneous compartment and well-stirred compartment become redundant. In this paper, I plan to restrict myself to compartments as defined above; therefore I shall consider only kinetics of order one with a constant rate w6x. With these hypotheses, if x Ž t . is the number of particles in a compartment, the differential equation d xrdt s yK x Ž t . q r Ž t .
Ž1.
holds, with K the constant rate of elimination from U
Corresponding author. 14767 Square Lake Trail N., Stillwater, MN 55082-9278, USA.
1043]6618r99r060471]08r$30.00r0
PURPOSE OF MODELING It has been said ad nauseam that the purpose of modeling is to test hypotheses w7]10x, yet frequently we see in the literature statements like ‘models of data’ w11x, meaning that experimental data are fitted to ‘abstract mathematical functions... that do not necessarily have any physical basis’; the so-called models of data may be useful for synthetically describing data observed in the past or to interpolate some values, but such an analysis has no scientific content per se. A classical example of ‘model of data’ is the Almagest; in it Ptolemy used abstract mathematical functions Žcycles and epicycles. to describe the movements of all known celestial bodies, obtaining a remarkably good fit; but the good fit did not help in interpreting the data or in predicting future observations. On the contrary, Aristarchus of Samos Ž; 310]230 BC. first, then more consistently Copernicus ŽAD 1473]1545., put forward the heliocentric hypothesis, thereby basing their computations not on the mere fitting of data Žin fact Copernicus fitting was not better than Ptolemy’s., but on a physical basis. In short, a compartmental equation like Ž1. implies the hypothesis that the particles present in the compartment leave it at a constant fractional rate, K; this may or may not be the case. There are instances where the particles present in a given space are eliminated with a non-linear process, or Q 1999 Academic Press
Pharmacological Research, Vol. 39, No. 6, 1999
472
with a linear process but not exponentially Žfor instant by diffusion., or by an exponential process with a non-constant fractional rate. Of course all results obtained with the application of equation Ž1., an apparently very general equation, are valid only within the validity of the stated hypotheses. Fitting the experimental data with a sum of exponential functions, by itself, does not guarantee the validity of the above hypotheses, and the determination of the parameters of the exponential functions is not scientifically meaningful if those parameters cannot be given a physical or physiological interpretation. Another example of dubious use of compartmental analysis is in the choice of the model. This choice is sometimes done according to some criteria like ‘identifiability’, as though an identifiable model were more ‘true’ than a non-identifiable one. The truth is that a ‘non-identifiable’ model may permit the determination of the range of some meaningful parameters, while with an ‘identifiable’ model you may compute the exact value of some parameters deprived of any physical or physiological meaning. I shall now try to show, with some examples, how to get the most from compartmental models.
DEGREES OF FREEDOM
A system of n compartments is completely described by nŽ n y 1. transfer rates and n turnover rates, for a total of n2 parameters. If one compartment is sampled after a bolus administration, we observe a sum of n exponential functions, therefore we know n exponents and n y 1 coefficients, for a total of 2 n y 1 parameters; thus a system of n compartments can be determined with Ž n y 1. 2 degrees of freedom. A system of two compartments can be determined with just 1 degree of freedom, with three compartments the degrees of freedom are 4, and with more compartments the number of degrees of freedom increases very quickly. Nevertheless, there is a considerable amount of meaningful information that can be extracted from such systems. This problem has been dealt with previously in some special cases, notably by Berman and Schoenfeld w12x and by Mordenti and Rescigno w13x. Here I shall show a general solution in the case of two and three compartments, leaving the solution for more compartments to the imagination of the reader.
d x1 s yK 1 x 1 q k 21 x 2 dt d x2 s qk 12 x 1 y K 2 x 2 , dt
Ž2.
where x 1Ž t ., x 2 Ž t . are the amount of drug at time t in each compartment, the capital K s are the turnover rates of the compartments, and the small ks are the transfer rates between compartments. Obviously there are the restrictions 0 F k 12 F K 1 , 0 F k 21 F K 2 .
Ž3.
For simplicity we suppose that the dose D of drug is administered into the first compartment as a bolus at time t s 0, and sampled in that same compartment; we thus have the initial conditions x 1Ž 0 . s D, x 2 Ž 0 . s 0. What we measure is not the amount present in a compartment, but its concentration, i.e. c1Ž t .. At this point it is very convenient to manipulate the above equations with the operational notation first developed in 1959 w14x, and in 1972 applied to compartmental analysis w6x. The interested reader may get more information from the two above quoted references. In Appendix A, I have listed only a number of rules as they will be used in the following pages. Equations Ž2. can be written, using rule 1 Žsee Appendix A., s x 1 4 y x 1Ž 0 . s yK 1 x 1 4 q k 21 x 2 4 s x 2 4 y x 2 Ž 0 . s qk 12 x 1 4 y K 2 x 2 4 and incorporating the initial conditions, q Ž s q K 1 . x 1 4 y k 21 x 2 4 s D yk 12 x 1 4 q Ž s q K 2 . x 2 4 s 0.
Ž4.
Using ordinary algebra, from equations Ž4. we have the solution x1 4 s
s q p1 , s 2 q q1 s q q 2
Ž5.
where TWO COMPARTMENTS
The differential equations describing a system of two compartments are
p1 s K 2 ,
q1 s K 1 q K 2 ,
q2 s K 1 K 2 y k 12 k 21 . Ž 6 .
On the other hand, we measure the concentration of the drug, and from the experimental data we get a
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sum of two exponential functions c1Ž t . s A1 eyl 1 t q A 2 ey l 2 t ,
Ž7.
istered as a bolus at time t s 0 is given by the integral from 0 to ` of x 1Ž t ., therefore, using rule 3 on equation Ž5., T1 s p1rq2
and using rule 2, c1 4 s s
A1 A2 q s q l1 s q l2 Ž A1 q A 2 . s q A1 l2 q A 2 l1 s 2 q Ž l1 q l 2 . s q l1 l 2
Ž8.
We know that c1Ž0. s DrV1 , where V1 is the volume of the first compartment, therefore from equation Ž6. we get V1 s
D , A1 q A 2
p1 s
A1 l2 q A 2 l1 , A1 q A 2
As shown by Rescigno and Gurpide w16x, the exit time from the sampling compartment is given by the ratio w H0` t ? x 1Ž t . dt xrw H0` x 1Ž t . dt x, therefore using rules 3 and 4, V1 s
Following Mordenti and Rescigno w13x, the exit time from the system of both compartments can be computed with the formula `
V 12 s
q1 s l1 q l2 ,
K 1 s q1 y p 1 ,
K 2 s p1 ,
k 12 k 21 s p1Ž q1 y p1 . y q2 .
k 21 s
p 1 Ž q1 y p 1 . y q 2 a
p1 q a . q2
q1 q 1 y F V 12 F 1 . q2 p1 q2
p 1 Ž q1 y p 1 . y q 2 F p1 , a
q2 F a F q1 y p 1 . p1
, w k 10 x 1Ž t . q k 20 x 2 Ž t .x dt
Using inequalities Ž9.,
The dilution factor u , as shown by Mordenti and Rescigno w13x, is
therefore, q1 y p 1 y
`
V 12 s
and using conditions Ž3., 0 F a F q1 y p 1 , 0 F
t ? w k 10 x 1Ž t . q k 20 x 2 Ž t .x dt
where k 10 s K 1 y k 12 and k 20 s K 2 y k 21 are not known exactly; but we can compute a range for this parameter. Using the arbitrary parameter a and the previous results we get
Introducing the arbitrary parameter a we have k 12 s a ,
H0
H0
q2 s l1 l2 . Now from equation Ž5. we can compute
q1 1 y . q2 p1
us
Ž9.
In summary, K 1 s q1 y p 1 , K 2 s p 1 , q q1 y p1 y 2 F k 12 F q1 y p1 , p1 q2 p1 y F k 21 F p1 . q1 y p 1
DETERMINATION OF OTHER PARAMETERS Besides turnover times and transfer times, there are many other important parameters that can be computed from the data available. It has been shown w13, 15x that the permanence time in the compartment where the drug was admin-
K 2 q k 12 , K2
therefore
us1q
a p1
and q1 q q y 22 F u F 1 . p1 p1 p1
EXAMPLE Suppose we gave an i.v. dose of 100 mg of a drug to a subject, and measured the concentration c1Ž t . s 33.5ey1 .0 t q 10.2ey4.0 t ,
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474
where the coefficients are measured in mg ly1 and the exponents in hy1 . We compute
d x1 s yK 1 x 1 q k 21 x 2 q k 31 x 3 dt d x2 s qk 12 x 1 y K 2 x 2 q k 32 x 3 dt d x3 s qk 13 x 1 q k 23 x 2 y K 3 x 3 dt
100 s 2.29 l 33.5q 10.2 33.5= 4.0q 10.2= 1.0 p1 s s 3.30 hy1 33.5q 10.2 V1 s
q1 s 1.0q 4.0s 5.0 hy1 q2 s 1.0= 4.0s 4.0 h
Ž 10.
with the restrictions
y2
ki j
and
G 0,
i , j s 1,2,3;
Sj ki j F K i , i
s 1,2,3
Ž 11.
y1
K 1 s 5.0y 3.30s 1.7 h K 2 s 3.30 hy1
and the initial conditions x 1Ž 0 . s D,
k 12 k 21 s 3.30= 1.7y 4.0s 1.61 hy2 Now we introduce the parameter a with the restrictions
x 2 Ž 0 . s x 3 Ž 0 . s 0.
Proceeding as before, we have the equations q Ž s q K 1 . x 1 4 y k 21 x 2 4 y k 31 x 3 4 s D
4.0 1.7y F a F 1.7, 3.30
yk 12 x 1 4 q Ž s q K 2 . x 2 4 y k 32 x 3 4 s 0 yk 13 x 1 4 y k 23 x 2 4 q Ž s q K 3 . x 3 4 s 0
therefore 0.49F k 12 F 1.7
and using ordinary algebra,
0.95F k 21 F 3.29 x1 4 s
We can also compute the permanence time in the first compartment, 3.30 T1 s s 0.82 h, 4.0 the exit time from the first compartment, V1 s
5.0 1 y s 0.95 h, 4.0 3.3
the exit time from the system, 5.0 1 5.0 y F V 12 F 4.0 3.3 4.0 thence
s 2 q p1 s q p 2 s 3 q q1 s 2 q q 2 s q q 3
Ž 12.
where p1 s K 2 q K 3 , p 2 s K 2 K 3 y k 23 k 32 , q1 s K 1 q K 2 q K 3 , q2 s K 1 K 2 y k 12 k 21 q K 1 K 3 y k 13 k 31 qK 2 K 3 y k 23 k 32 , q3 s K 1 K 2 K 3 y K 1 k 23 k 32 y K 2 k 13 k 31 y K 3 k 12 k 21 yk 12 k 23 k 31 y k 13 k 32 k 21 .
Ž 13.
Now we can compute 0.95F V 12 F 1.25,
and finally the dilution factor 5.0 4.0 5.0 y FuF 2 3.30 3.30 3.30 thence
K 1 s q1 y p 1 ,
K2 s a ,
K 3 s p1 y a ,
where a is an arbitrary parameter; we are left with equations Ž14.: p 2 s a Ž p1 y a . y k 23 k 32 q2 s p1Ž q1 y p1 . y k 12 k 21 y k 13 k 31 q p 2
1.15F u F 1.52.
q3 s a Ž p1 y a .Ž q1 y p1 . y Ž q1 y p1 . k 23 k 32 Ž 14. ya k 13 k 31 y Ž p1 y a . k 12 k 21
THREE COMPARTMENTS The differential equations describing a system of three compartments are
yk 12 k 23 k 31 y k 13 k 32 k 21 . These are three equations in the six unknown transfer rates; the solution depends on three more arbitrary parameters besides a .
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From the first of the above equations we get
then for the products of the transfer rates
k 23 k 32 s a Ž p1 y a . y p 2 , but from Ž11. it is necessary that
k 12 k 21 s
K 2 Ž q2 y K 1 K 2 y K 2 K 3 . y q3 , K3 y K2
k 13 k 31 s
q3 y K 3 Ž q2 y K 1 K 3 y K 2 K 3 . . K3 y K2
k 23 k 32 G 0,
To determine the transfer rates we introduce two arbitrary parameters b and g , such that
therefore
a Ž p1 y a . y p 2 G 0,
k 12 s b , k 21 s k 12 k 21rb , k 13 s g , k 31 s k 13 k 31rg .
and solving this inequality we get p1 y 2
(ž
p1 2
2
/ yp
2
FaF
p1 q 2
p1 2
(ž
2
/ yp
2
.
Conditions Ž11. require that
Ž 15.
b ) 0, g ) 0, b q g F K 1 , k 12 k 21rb F K 2 , k 13 k 31rg F K 3 ,
For a equal to its lowest or largest admissible value we have k 23 k 32 s 0,
therefore
while the largest possible value for k 23 k 32 is given by
as
p1 , 2
k 23 k 32 s
ž p2 / y p . 1
2
b q g F K1 b G k 12 k 21rK 2 g G k 13 k 31rK 3
2
The large number of degrees of freedom in the determination of the transfer rates may seem forbidding, but there is still a large amount of information that we can get from the available data. We start by considering two particular cases, i.e. a catenary system and a mammillary system.
MAMMILLARY SYSTEM
The three above inequalities define the triangle ABC Žsee Fig. 1. whose points correspond to acceptable values of the parameters b and g . Observe that along line AB, b assumes its minimum value, therefore k 21 s K 2 and there is no loss from the second compartment. Along line BC, b q g s K 1 , therefore k 12 q k 13 s K 1 and there is no loss from the first compartment. Along line CA, g assumes its minimum value, therefore k 31 s K 3 and there is no loss from the third compartment. All
In a mammillary system w6x, k 23 s k 32 s 0, therefore equations Ž13. become p1 s K 2 q K 3 , p2 s K 2 K 3 , q1 s K 1 q K 2 q K 3 ,
Ž 16.
q2 s K 1 K 2 y k 12 k 21 q K 1 K 3 y k 13 k 31 q K 2 K 3 , q3 s K 1 K 2 K 3 y K 2 k 13 k 31 y K 3 k 12 k 21 , and we solve immediately for the turnover rates K 1 s q1 y p 1 , K2 s
p1 y 2
K3 s
p1 q 2
(ž (ž
p1 2 p1 2
2
/ yp 2
/
2
,
y p2 ,
Fig. 1. Mammillary model of a three-compartment system. Range of parameters b and g that define physically realizable values of the model parameters.
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points inside triangle ABC correspond to intermediate states.
CATENARY SYSTEM In a catenary system w6x, k 13 s k 31 s 0, therefore equations Ž13. become p1 s K 2 q K 3 , p 2 s K 2 K 3 y k 23 k 32 , q1 s K 1 q K 2 q K 3 , q2 s K 1 K 2 y k 12 k 21 q K 1 K 3 q K 2 K 3 y k 23 k 32 , q3 s K 1 K 2 K 3 y K 1 k 23 k 32 y K 3 k 12 k 21 , Fig. 2. Catenary model of a three-compartment system. Range of parameters b and g that define physically realizable values of the model parameters.
and we can compute immediately K 1 s q1 y p 1 while from the remaining four equations K 2 q K 3 s p1 ,
EXAMPLE Suppose we have found that
K 2 K 3 y k 23 k 32 s p 2 , x1 4 s
k 12 k 21 q k 23 k 32 s K 1 K 2 q K 1 K 3 q K 2 K 3 y q2 , K 3 k 12 k 21 q K 1 k 23 k 32 s K 1 K 2 K 3 y q3 , we can compute the turnover rates K 2 and K 3 and the products of the transfer rates k 12 k 21 and k 23 k 32 . To determine the individual transfer rates we introduce the indeterminate parameters b and g , such that
With the hypothesis that the system is mammillary we can compute the turnover rates K 1 s 2.0,
K 2 s 1.0,
k 12 k 21 s 0.60,
k 12 s k 12 k 21rb ,
K 3 s 3.5
and the products
k 21 s b ,
k 13 k 31 s 1.70;
then with the two arbitrary parameters b and g we have
k 23 s g ,
k 12 s b ,
k 32 s k 23 k 32rg ,
k 21 s 0.60rb ,
k 13 s g ,
k 31 s 1.70rg
with the conditions
with the conditions k 12 k 21rb F K 1 ,
s 2 q 4.5s q 3.5 . s 3 q 6.5s 2 q 10.5s q 4.25
b q g F K2 ,
k 23 k 32rg F K 3
The three above inequalities define the triangle ABC Žsee Fig. 2. whose points correspond to acceptable values of the parameters b and g . Observe that along line AB, b assumes its minimum value, therefore k 12 s K 1 and there is no loss from the first compartment. Along line BC, b q g s K 2 , therefore k 21 q k 23 s K 2 and there is no loss from the second compartment. Along line CA, g assumes its minimum value, therefore k 32 s K 3 and there is no loss from the third compartment. All points inside triangle ABC correspond to intermediate states.
b q g F 2.0,
b G 0.60, g G 0.486.
If we accept the alternative hypothesis that the system is catenary, we can compute K 1 s 2.0 and we are left with the four equations K 2 q K 3 s 4.5, K 2 K 3 y k 23 k 32 s 3.5, k 12 k 21 q k 23 k 32 y 2 K 2 y 2 K 3 y K 2 K 3 s y10.5, K 3 k 12 k 21 q 2 k 23 k 32 y 2 K 2 K 3 s y4.25,
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whose solution is K 2 s 3.125,
K 3 s 1.373,
k 12 k 21 s 2.0,
k 23 k 32 s 0.80,
477
in fact when a is in the interval 2.25]3.5, we have the same solution with the second compartment switched with the third one. Eliminating k 13 k 31 from the above equations we have
then we eliminate K 2 q K 3 and we get k 12 k 21 s 2.0,
K 3 s 1.375,
K 2 s 3.125,
and finally k 23 k 32 s 0.80. Now we introduce two parameters b and g , such that k 21 s b ,
k 12 s 2.0rb ,
k 23 s g ,
k 32 s 0.80rg .
k 12 k 21 s
2 Ž 1.375y a . y k 12 k 23 k 31 y k 13 k 32 k 21 ; 2 Ž 2.25y a .
for this expression to be non-negative it is necessary that
a F 1.375. We conclude that 1.0F a F 1.375; our parameters are therefore K 1 s 2, 1.0F K 2 F 1.375, 3.125F K 3 F 3.5, 0 F k 23 k 32 F 0.7969.
Inequalities Ž11. require that
b G 1.0,
b q g F 3.125, g G 0.604.
General case Without any special hypothesis on the transfer rates, equations Ž14. become 3.5s a Ž 4.5y a . y k 23 k 32 ,
Even though we were not able to determine the exact value of the turnover rates, we were able to restrict their range to a reasonably narrow interval. Additionally, we can compute the permanence time in the first compartment,
10.5s 4.5 Ž 6.5y 4.5. y k 12 k 21 y k 13 k 31 q 3.5,
T1 s
4.25s a Ž 4.5y a .Ž 6.5y 4.5. y Ž 6.5y 4.5. k 23 k 32 ya k 13 k 31 y Ž 4.5y a . k 12 k 21
the exit time from the first compartment,
yk 12 k 23 k 31 y k 13 k 32 k 21 ,
V1 s
thence k 23 k 32 s ya 2 q 4.5 a y 3.5,
10.5 4.5 10.5 y F V 12 F 4.25 3.5 4.25 thence
qk 13 k 32 k 21 s y2 a q 9.0 a y 4.25, 2
and finally
10.5 4.5 y s 1.714 h, 3.5 3.5
the exit time from the system,
k 12 k 21 q k 13 k 31 s 2.0, Ž 4.5y a . k 12 k 21 q a k 13 k 31 q 2 k 23 k 32 q k 12 k 23 k 31
3.5 s 0.824 h, 4.25
1.18F V 12 F 2.47, and finally the dilution factor
k 23 k 32 s ya 2 q 4.5 a y 3.5,
10.5 4.25? 4.5 10.5 y FuF 2 3.5 3.5 Ž 3.5.
k 12 k 21 q k 13 k 31 s 2.0, Ž 4.5y a . k 12 k 21 q a k 13 k 31 q k 12 k 23 k 31
thence
qk 13 k 32 k 21 s 2.75. We have three equations with six unknowns; they can be solved with three additional degrees of freedom, besides the parameter a ; we already know, from inequality Ž15., that 1.0F a F 3.5. Now observe that without any loss of generality we can restrict the range of a to the interval 1.0F a F 2.25;
1.44F u F 3.00.
APPENDIX A Rule 1: Derivative of a function
½ ddtf 5 s s f 4 y f Ž0. .
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Rule 2: Exponential function aeyl t 4 s
a . sql
Rule 3: Integral of a function `
H0
f Ž t . dt s lim f 4 . sª0
Rule 4: First moment of a function `
d f lim H0 t ? f Ž t .dt s sª0 ds
4
.
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5. Rescigno A, Segre G. Drug and Tracer Kinetics. Waltham, Massachusetts: Blaisdell, 1966. 6. Rescigno A, Beck JS. Compartments. In: Rosen R, Ed. Foundations of Mathematical Biology. New York: Academic Press, 1972: 255]322. 7. Nooney GC. Mathematical models, reality and results. J Theor Biol 1965; 9: 239]52. 8. Bergner P-EW. Tracer theory: a review. Isot Radiat Technol 1966; 3: 245]62. 9. Beck JS, Rescigno A. Calcium kinetics: the philosophy and practice of science. Phys Med Biol 1970; 15: 566]7. 10. Rescigno A, Beck JS. The use and abuse of models. J Pharmacokinet Biopharm 1987; 15: 327]40. 11. DiStefano JJ, Landaw EM. Multiexponential, multicompartmental, and noncompartmental modeling. I. Methodological limitations and physiological interpretations. Am J Physiol 1984; 246: R651]64. 12. Berman M, Schoenfeld R. Invariants in experimental data on linear kinetics and the formulation of models. J Appl Phys 1956; 27: 1361]70. 13. Mordenti J, Rescigno A. Estimation of permanence time, exit time, dilution factor, and steady-state volume of distribution. Pharm Res 1992; 9: 17]25. 14. Mikusinski J. Operational Calculus. London: Pergamon Press, 1959. 15. Rescigno A, Thakur AK, Brill AB, Mariani G. Tracer kinetics: a proposal for unified symbols and nomenclature. Phys Med Biol 1990; 35: 449]65. 16. Rescigno A, Gurpide E. Estimation of average times of residence, recycle, and interconversion of bloodborne compounds using tracer methods. J Clin Endocrinol Metab 1973; 36: 263]76.